ICARUS 99, 1--14 (1992)
Modeling the Martian Seasonal C02 Cycle
1. Fitting the Viking Lander Pressure Curves
STEPHEN E . WOOD AND DAVID A . PAIGE
Department of Earth and Space Sciences, University of California, Los Angeles, Los Angeles, California 90024
Received January 29, 1992; revised May 19, 1992
volves a large fraction of the mass of the Martian atmosphere, as well as a large fraction of the surface of the
planet. Any realistic model for the polar caps and atmosphere of Mars should at least be able to reproduce the
seasonal variations exhibited by these data before attempting to predict the behavior of the Martian CO2 cycle
during other climatic epochs.
The first serious attempt to reproduce the Viking pressure curves for a complete annual cycle was by James and
North (1982). They employed a one-dimensional model
of the North-Coakley (North et al. 1981) type, which
solved for the surface heat balance as a function of latitude. All quantities were diurnally and zonally averaged,
with an integration period of 1/200 of a Martian year. The
surface was modeled as a single soil layer with a fixed
heat capacity, but true subsurface heat conduction was
neglected. James and North were not able to simultaneously match the amplitudes and the relative depths of the
minima of the seasonal pressure variations observed by
Viking using this simple, one-dimensional model. They
were, however, able to obtain a substantially better fit
Press, Inc.
using a more complex, two-dimensional model that included the expected effects of global dust storms, polar
hood clouds, and other atmospheric phenomena.
In this paper, we present the results of our own efforts
INTRODUCTION
to match the Viking Lander pressure data using a straightThe atmospheric surface pressure records acquired at forward diurnal and seasonal thermal model which inthe two Viking Lander sites are extremely valuable for cludes an accurate treatment of the potential effects of
studying the Martian climate. These data show short heat conduction. The model is similar in function to the
time-scale fluctuations due to dynamical phenomena in original model developed in 1966 by Leighton and Murray
the Martian atmosphere, seasonal time-scale variations (1966), which was the first to demonstrate that the advance
due to the condensation and sublimation of carbon dioxide and retreat of the seasonal polar caps on Mars can be
frost in the Martian north and south polar regions, and explained by the condensation and sublimation of CO 2
interannual variations due to year-to-year differences in frost in solid-vapor equilibrium with CO2 gas in the Marthe Martian weather and in the Martian seasonal CO2 tian atmosphere. We find that this type of model can
cycle (Hess et al. 1979, Ryan and Henry 1979, Tillman et accurately simulate most of the major processes responsible for the net heat balance of the Martian polar caps using
al. 1979, Leovy 1981, Leovy et al. 1985, Barnes 1980,
1981, and TiUman 1988) (Fig. 1). This seasonal cycle in- a minimum number of input parameters. Its simplicity
A diurnal and seasonal thermal model is used to simulate the
seasonal exchange of carbon dioxide between the atmosphere and
polar caps of Mars. Surface CO2 frost condensation and sublimation rates are determined by the net effects of radiation, latent
heat, and heat conduction in subsurface soil layers. We show
that this model can successfully reproduce the measured seasonal
pressure variations at the Viking lander 1 site using constant
values for frost albedo and emissivity in each hemisphere. An exact
treatment of heat conduction is found to have important effects,
as our best-fit CO2 frost albedos and emissivities are not unique,
but depend on the value of soil thermal inertia assumed in each
hemisphere. We find that Martian seasonal pressure variations are
primarily due to frost condensation and sublimation in the 55 o to
70 ° latitude regions in both hemispheres. The observed retreat of
the north and south seasonal polar caps can also be matched closely
by this model, but no combination of best-fit frost albedos and
emissivities was consistent with the stability of permanent CO2
deposits at either pole. The fact that this relatively simple model
can provide such a good fit to the Viking Lander 1 pressure curve
makes it a useful platform for studying the Martian CO2 cycle over
seasonal, interannual, and climatic timescales. © 1992 Academic
0019-1035/92 $5.00
Copyright © 1992 by Academic Press, Inc.
All rights of reproduction in any form reserved.
2
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FIG. 1. Daily average a t m o s p h e r i c surface p r e s s u r e s on Mars m e a s u r e d by Viking L a n d e r 1 (VL1) and Viking L a n d e r 2 (VL2). T h e s e are part
o f the reduced Viking L a n d e r P r e s s u r e D a t a s e t provided by the Planetary Data S y s t e m (see References). V L I recorded p r e s s u r e s from July 1976
to S e p t e m b e r 1982. V L 2 recorded p r e s s u r e s from S e p t e m b e r 1976 to A u g u s t 1979. VL2 p r e s s u r e s are generally higher d u e to its lower elevation.
Viking years are defined here to begin at L s 0 (vernal equinox in the North), and the s y m b o l s c o r r e s p o n d to the y e a r s in w h i c h the data were taken.
Only daily average p r e s s u r e s b a s e d on m e a s u r e m e n t s with a m a x i m u m time gap of less t h a n 6 hr are s h o w n . T h e initiation and d e c a y p h a s e s of
the 1977A and 1977B global dust s t o r m s that occurred in the first Viking year are also indicated.
allows us to model the whole planet with high spatial and
temporal resolution and still investigate a wide range of
parameter values to find the best possible fit to the pressure data. F u r t h e r m o r e , by varying these same input parameters, this model can be used to simulate the net effects of other potentially important processes which are
not included explicitly.
MARS THERMAL
MODEL
We have constructed a Mars thermal model similar to
those used in previous studies (Leighton and Murray 1966,
Kieffer et al. 1977, Clifford and Bartels 1986, Paige 1992).
Radiative, conductive, and latent heat balance is maintained at the surface, and the one-dimensional heat conduction equation is solved numerically for each subsurface layer.
The instantaneous surface energy balance equation is
S(1 - A) cos i - F.o'T4 + k d T + L d m = O,
dz
dt
(i)
where S is the normal solar flux at the current M a r s - S u n
distance; A is the bare ground or frost albedo; i is the solar
incidence angle; ~ is the bare ground or frost emissivity;
T is the bare ground surface temperature or the temperature o f surface CO2 frost when present; k is the thermal
conductivity o f the soil; d T / d z is the vertical temperature
gradient evaluated at the surface with z positive downward; L is the latent heat o f sublimation o f CO2 frost; d m /
dt is the time rate of change of the mass per unit area of
the CO2 frost deposit. These terms are calculated every
1/36 of a Martian day, which corresponds to a time step
of 41 min. Forty-two latitude bands were used, with a
FITTING THE VIKING LANDERPRESSURE CURVES
resolution of 2° near the poles and 5° at lower latitudes,
with flat topography over the entire globe. The subsurface
is divided into 40 layers of increasing thickness down to
several annual skin depths. Subsurface temperatures are
initialized to the estimated annual average surface temperature for that latitude. The one-dimensional heat conduction equation
dT
dt
-
k d2T
pc dz 2
(2)
is solved numerically, and the temperature of each layer
is updated at time intervals which also increase with
depth, but are consistent with a stable, nonoscillatory
solution. We used l06 J m -3 K -1 for the product of the
density p and the heat capacity c for all layers. The thermal
conductivity is cast in terms of the thermal inertia, I =
X/kpc, which is proportional to the heat flux in and out of
the soil. The deep layers affected only by the seasonal
thermal wave are assumed to have the same thermal inertias as layers near the surface.
The atmosphere used in our model is composed of CO2
and a small percentage (7.9 kg m -l) of noncondensible
gases, and is assumed to be transparent to all radiation.
Since atmospheric CO 2 is in vapor equilibrium with the
seasonal frost deposits, its partial pressure determines the
frost-point temperature by the solid-vapor equilibrium
relation (Fanale et al. 1982). This frost-point temperature
is maintained wherever any frost is on the surface by the
condensation/sublimation of enough CO2 to make up any
deficit/excess in the energy balance through the release/
absorption of latent heat. Since the total mass of CO2 in
the cap-atmosphere system is fixed, subtracting the total
CO2 condensed on the surface gives the atmospheric
mass. Using a gravitational acceleration of 3.72 m sec -2,
the atmospheric pressure is calculated at each time step
and then used to recalculate the frost-point temperature.
These adjustments can be made instantaneously because
the rate at which these pressure changes are propagated
is determined by the sound speed, so a disturbance can
travel l0 ° of latitude in one time step and the circumference of Mars in one day.
FITTING THE VIKING LANDER PRESSURE CURVES
A. Procedure
In order to evaluate the success of our attempts to
match the data, we used a smooth reference curve based
on the daily average surface pressures measured by Viking Lander 1 (VL1) (22.48 ° N, 47.8 ° W) (Tillman 1989).
Some of the sols contained one or more data gaps lasting
several hours. For our reference curve, we used daily
average pressures only from those sols with a maximum
time gap of less than 6 hr. The VL1 data show less suscep-
3
tibility to local weather disturbances than those obtained
by Viking Lander 2 (VL2) (47.96 ° N, 225.59 ° W); therefore
we decided to use only VL1 data as a basis for comparison. As can be seen in Fig. 1, both Viking Lander sites
experienced an extended period of elevated daily average
pressures after the beginning of the 1977B global dust
storm. Although it is more obvious in the VL2 data, the
deviation during this period in the first year of VL1 data
is significant when compared with the second and third
years, when no large global dust storms were observed.
For this reason, VL1 first year data were not used to
constrain the model, although the rest of the seasonal
variation is virtually identical in all 4 years. VL1 fourth
year data were not obtained for a complete annual cycle,
so only the second and third years were used to make our
reference curve. Since these pressure records had many
gaps, we combined the data from both years and averaged
pressures measured on the same day. These were then
smoothed by averaging all of the pressures within 7 sols
of each point and filling small gaps of less than 15 sols by
interpolation. This removed the high frequency
"weather" components to create a nearly continuous
curve. There was only one good daily average pressure
value between L s 270 and 291, so this period was not
interpolated. Figure 2 shows the resulting reference curve
in comparison with the data.
Before beginning a systematic search of parameter
space to find the best fit to the Viking Lander pressure
data, we first defined a set of ground rules. After some
initial experimentation to investigate the model's sensitivity to the parameters in the heat balance equation, we
decided to independently vary six free parameters; the
frost albedo, the frost emissivity, and the soil thermal
inertia, in both hemispheres. We also decided to keep the
chosen values for each parameter constant with time for
each model run. Our experimentation showed that atmospheric pressures and the sizes of the seasonal polar caps
were extremely sensitive to the assumed albedo and emissivity of the CO 2 frost, but that changing the bare soil
albedo and emissivity within reasonable limits (Palluconi
and Kieffer 1981) had little effect on the CO2 cycle, so
they were fixed at 0.25 and 0.95, respectively. It should
be noted that bare soil radiative properties and thermal
inertia in our model are only relevant for latitudes at which
CO z condensation occurs.
In our attempts to fit the Viking pressure curve, we
investigated a large number of combinations of CO 2 frost
albedos and emissivities between 0.30 and 1.00 for ten
different thermal inertias between 42 and 1674 J m -2
sec-l/2 K-I (MKS) in each hemisphere. The best-fit albedos and emissivities were determined to an accuracy of
1% absolute by using increasing resolution to focus in on
the region of parameter space that gave the best fit for a
given pair of thermal inertias. We allowed at least four
4
WOOD AND PAIGE
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FIG. 2. Comparison of the reference curve and the VL1 second and third year data (November 1977-August 1981) which were smoothed and
interpolated to create it. This reference curve was used to evaluate the standard deviation of model-generated pressure curves.
Mars years for stabilization, then output the calculated
daily a v e r a g e surface pressures and seasonal polar cap
boundaries for a c o m p l e t e annual cycle. Cases that
yielded p e r m a n e n t polar caps did not converge completely
after 4 years since the residual deposit accumulated more
frost each year. Although the annual average pressures
continued to fall, the shapes of the seasonal pressure
variations remained the same b e c a u s e frost-point temperature depends only v e r y weakly on v a p o r pressure. H o w ever, these cases n e v e r resulted in a close match to the
VL1 data (see Discussion and Interpretation, Section B).
Surface pressures are calculated by our model at the altitude of the polar caps, which c o r r e s p o n d s to the zero
reference altitude. Since our model has n o topography,
our output daily average pressures were adjusted to the
m e a s u r e d VL1 altitude of - 1 . 5 k m (Michael et al. 1976,
Christensen 1975) using a c o n s t a n t scale height of 11.5 km
(Seiff and Kirk 1977). We found that slightly changing the
total mass of CO2 in the s y s t e m would raise or lower the
annual a v e r a g e pressures, but it had very little effect on
the shapes o f the curves or the m a s s of the polar caps,
again b e c a u s e it resulted in only a small change in frostpoint t e m p e r a t u r e . F o r all of our model runs, we kept the
total m a s s of CO2 fixed at 210 kg m -2, and then added or
subtracted a small, constant amount of pressure to or
from the model-calculated curves to give annual average
pressures that agreed with the observations. We ranked
the quality of each c u r v e ' s fit to the reference curve by
least-squares analysis. The cases with the lowest standard
deviation were run again, if n e c e s s a r y , using a total mass
of CO2 which resulted in the correct annual average pressure. The best-fit albedos and emissivities were then found
by checking values differing by 0.01 f r o m the original set
to see if a bett~r fit could be obtained using a self-consistent total m a s s of CO2.
B. Results
We were able to obtain excellent fits for a wide range
of model input p a r a m e t e r s , with standard deviations of
less than 10 Pa c o m p a r e d to an a v e r a g e pressure o f 800
Pa. Table I lists best-fit p a r a m e t e r values and standard
deviations for several representative cases. Figures 3, 4,
and 5 show the pressure curves and seasonal polar cap
boundaries generated by the model using these same par a m e t e r values. In general, the quality of the fit increased
with thermal inertia in the north, but the lowest inertias
w o r k e d best in the south.
E x a m i n a t i o n of our results showed that if the p a r a m e t e r
values in one h e m i s p h e r e are held constant, then the bestfit frost albedos and emissivities in the other are linearly
proportional to the thermal inertia. If we take the best-fit
values corresponding to a thermal inertia of 1340 (MKS)
in the north and 167 (MKS) in the south as a reference
FITTING THE VIKING LANDER PRESSURE CURVES
TABLE I
Best-Fit P a r a m e t e r Values
Thermal
inertia
(MKS)
Frost
albedo
Frost
emissivity
Total CO2
in system
(kg/m 2)
Standard
deviation
(Pa)
North
South
1591
42
0.59
0.59
1.00
0.63
207
3.4
North
South
670
167
0.70
0.54
0.69
0.71
210
4.7
North
South
272
272
0.77
0.54
0.55
0.78
214
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the corresponding parameter values from Table I. The latitudes of the boundaries of the seasonal polar caps calculated for this case are shown in
comparison to Viking Orbiter observations (triangles) of the retreating cap edges in 1977 (James 1979, James et al. 1979).
6
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FIG. 4. Best fit pressure curve using thermal inertias of 670 and 167 (MKS) in the north and south, respectively. This case yielded a north
seasonal polar cap which matched the polar cap retreat data most closely.
case, and hold them fixed in one hemisphere at a time,
then we obtain the best-fit albedo and emissivity values,
and standard deviations shown in Figs. 6a and 6b. The
points at which the required emissivities reach 1.0 put
upper limits on thermal inertia and lower limits on frost
albedo in each hemisphere. As shown in Figs. 6a and 6b,
we were not able to obtain satisfactory fits to the pressure
curves for thermal inertias of greater than 1591 and 837
(MKS), and albedos of less than 0.60 and 0.39 in the north
and south, respectively.
C. Polar Cap Retreat Rates
The retreat rates of the seasonal polar caps observed
by the Viking Orbiters (James 1979, James et al. 1979)
were not used as criteria for determining best-fit parameter values, but it can be seen in Figs. 3-5 that the edges
of the northern and southern seasonal polar caps calculated for these best-fit (to the pressure curve) cases agree
well with these observations. The northern cap matches
the retreat data almost perfectly for the case that assumes
FITTING THE VIKING LANDER PRESSURE CURVES
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FIG. 5.
Best fit pressure curve using " a v e r a g e " Martian soil thermal inertia, 272 (MKS) in both hemispheres.
a thermal inertia of 670 (MKS) in the northern hemisphere
(Fig. 4). Our model-calculated polar cap boundaries correspond to the lowest latitudes with a daily average mass of
frost greater than zero. However, the depth and distribution of frost that must be present on the surface to b e
observed from orbit is not known.
DISCUSSION AND INTERPRETATION
The fact that a relatively simple diurnal and seasonal
thermal model can produce seasonal pressure variations
that are in excellent agreement with those observed at
the VL1 site brings up a number of interesting questions
regarding the processes responsible for these pressure
variations, and the "realism" of the model we have employed.
A. The Effects of Heat Conduction
The results of our study can leave little doubt that heat
conduction has important effects on the behavior of the
Martian seasonal polar caps. This is not surprising, given
the large seasonal temperature variations that are experienced at high latitudes. Our results in Fig. 6 show that the
8
WOOD AND PAIGE
Northern Hemisphere
B e s t Fit V a l u e s
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sonal polar caps. We also found that the best-fit cases that
assumed high thermal inertias in the south were not able
to match the well-observed and highly consistent pressure
drop from L~ 60 to 150 as well as cases that assumed low
thermal inertias. In fact, cases that assumed a value of
even less than 42 (MKS) in the south worked best.
At the present time, we have no definitive information
concerning the thermal inertias o f Martian subsurface materials to seasonal skin depths. The fact that our best fits
to the VL1 pressure curves require higher thermal inertias
in the north than in the south is consistent with recently
completed high latitude thermal maps to diurnal skin
depths, which show average apparent inertias of - 2 5 0
(MKS) in the south polar region and - 450 (MKS) in the
north (Keegan e t a l . 1991, Paige and Keegan 1991). Actual
effective thermal inertias may be lower than indicated by
these measurements due to the effects of atmospheric
radiation (Haberle and Jakosky 1990), or higher due to the
possible presence of high thermal inertia water ice below
the surface (Paige 1992).
Our results also indicate that a reasonably exact treatment of the heat conduction process, including the diurnal
cycle, may be required (see Section C below). As a test,
we compared our model with James and N o r t h ' s (1982)
one-dimensional model by setting the thermal inertia
equal to 42 (MKS) and including a depth-dependent frost
albedo as they did, but neglected the small effects of
diffusive meridional heat transport. Our model produced
a much different curve (Fig. 7), even using the same frost
parameters.
soo
Soil Thermal
i000
1200
1400
isoo
Inertia (MKS)
FIG. 6. (a) Northern hemisphere and (b) southern hemisphere bestfit values of CO2 frost albedo and emissivity as a function of thermal
inertia. The standard deviation of the pressure curve generated by this
combination is plotted as a o- according to the scale on the right. These
values are slightly different than those for the same thermal inertia
combinations in Table I since parameters in one hemisphere at a time
were held fixed (solid symbols).
best-fit values for frost albedo and frost emissivity are
strongly dependent on assumptions concerning high latitude soil thermal inertias. The dependence is linear because seasonal subsurface heat storage rates are directly
proportional to thermal inertia.
We found that changes in thermal inertia and emissivity
each had different effects on c o m p u t e d pressure decreases
during the polar night seasons. Changing thermal inertia
maintained the slopes of these decreases, but tended to
move them forward or backward in time, whereas changing emissivity affected the slopes. Model cases that assumed higher thermal inertias in the northern hemisphere
were better able to fit the timing o f the decrease in atmospheric pressure observed from L s 250 to 360 because
warm subsurface temperatures tended to delay the onset
of CO2 frost condensation. When higher thermal inertias
were assumed, we required higher emissivities to fit the
L a n d e r pressure data because the higher thermal inertia
resulted in more frost sublimation at the base of the sea-
B. Permanent
Polar
Caps
Understanding the physical processes responsible for
the stability of permanent caps is extremely important for
Martian climate studies. Analysis of the Viking Orbiter
observations have given clear evidence for the presence
of a residual CO2 frost deposit at the south polar cap at
least during the first Viking year (Kieffer 1979, James
1979, Paige and Ingersoll 1985). Paige and Ingersoll's
(1985) annual radiation balance measurements are consistent with the south residual COz cap maintaining net heat
balance o v e r a complete annual cycle. In our study, we
found no combination of parameter values that could simultaneously fit the seasonal pressure variations at the
VL1 site and yield permanent CO2 caps at either pole.
Figure 8 compares our best-fit frost albedos and emissivities with the range of values that would permit the survival
of surface frost throughout summer at either pole. The
minimum albedos were determined by computing the annual average of Eq. (1) at the poles, assuming no heat
conduction to the subsurface and no net frost accumulation over the annual cycle. If the net effects of heat
conduction were included, then the critical albedos required for year-long stability would be higher. Jakosky
FITTING THE VIKING LANDER PRESSURE CURVES
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FIG. 7. Comparison of the best fit to the VLI pressure data of James and North's (1982) diurnally averaged one-dimensional model and the
results of our diurnal thermal model using the same frost parameters: an aibedo of 0.75 and an emissivity of 0.57 in both hemispheres. To simulate
the lack of heat conduction in their model, a thermal inertia of 42 (MKS) was used for the soil in each hemisphere. The seasonal polar cap boundaries
were calculated by our model.
a n d H a b e r l e (1990) h a v e s u g g e s t e d t h a t t h e M a r t i a n r e s i d ual polar caps can jump between two stable states; either
b e i n g c o v e r e d b y CO2 f r o s t all y e a r , o r h a v i n g t h e u n d e r l y i n g s u r f a c e e x p o s e d in t h e s u m m e r , d e p e n d i n g o n t h e
a m o u n t o f h e a t c o n d u c t e d i n t o t h e u n d e r l y i n g soil d u r i n g
previous years. However, the absence of residual polar
c a p s in o u r b e s t - f i t m o d e l s is n o t d u e t o t h i s e f f e c t . E v e n
i f w e i n i t i a t e d o u r m o d e l w i t h s m a l l r e s i d u a l CO2 c a p s at
both poles and cold soil temperatures below extending to
i n f i n i t e d e p t h s , t h e c a l c u l a t i o n s w o u l d still n o t r e s u l t in
stable residual polar caps because our best-fit albedos are
lower than even the minimum permanent cap albedos.
This result supports the idea that the present annual heat
b a l a n c e at t h e s o u t h r e s i d u a l p o l a r c a p is s o m e h o w u n i q u e
( K i e f f e r 1979, J a m e s a n d N o r t h 1982, P a i g e a n d I n g e r s o l l
1985).
10
WOOD AND PAIGE
1I
Permonent CO2
0.9
Stability ot Pole
0.8
0
(D
[]
[]
13
0.7
[]
<:
[]
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t,
0.6
0.5
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0.3
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North
•
South
0.4
0.5
•
0.6
0.7
0.8
0.9
F r o s t Emissivity
FIG. 8. Best fit albedos vs emissivities from Fig. 6. The shaded area
indicates the region of parameter space that yields permanent polar caps,
neglecting heat conduction.
C. Source of Seasonal Pressure Variation
We know that seasonal variations provide measures of
the rates of condensation and sublimation of CO 2 frost
over the entire globe of Mars, but it would also be useful
to know which regions of the planet are most involved
and when. To better interpret our results, we examined
the model output in greater detail to determine the spatial
and temporal distribution of seasonal frost condensation
and sublimation rates. Figure 9a shows a contour plot of
the diurnally averaged value of the L dm/dt term in Eq.
(1) for the best-fit case assuming a thermal inertia of 670
(MKS) in the north and 167 (MKS) in the south, using a
latitude resolution of 1°. Condensation rates increase with
latitude during the fall and winter seasons because higher
latitude regions receive less insolation. During the spring
and summer seasons, sublimation rates increase with
time, but are largely independent of latitude. Figure 9a
presents a somewhat misleading picture of which latitudes
have the greatest effects on seasonal pressure variations
because the results are not weighted by area. Figure 9b
includes the effect of scaling the calculated values ofL dm/
dt at each latitude by the area of its latitude band. The
results show that the seasonal exchange of CO2 between
caps and atmosphere is dominated by the behavior in the
55° to 70° latitude region in both hemispheres, and that
more than half of the fall and winter CO 2 condensation
occurs at latitudes that experience a diurnal insolation
cycle outside the polar night. Therefore, it appears that
our best-fit values for albedo and emissivity are most
relevant to the 55° to 70° latitude zones, and that modeling
the diurnal cycle can be important. This also implies that
activities occurring within the small north and south residual polar cap regions will not have large effects on seasonal pressure curves. However, when considering the
details of the seasonal pressure curves, the residual cap
areas could be of importance. The fact that our best-fit
model-calculated surface pressures tend to be lower than
the observations near summer solstice in the south may
be related to the failure of our model to accurately reproduce the terminal phases of the retreat of the south seasonal polar cap.
Looking at each hemisphere as a whole, it should be
noted that each seasonal polar cap does not control the
atmospheric pressure variation independently. It is true
that there are periods (summers) when only one seasonal
cap is active, and therefore only the frost properties of
that cap can have an effect on the atmospheric pressure.
These periods are useful for obtaining first order estimates
of the albedo and emissivity of each cap separately, but
there is no guarantee that these values will give the best
fit to the entire pressure curve. Even the durations of the
periods when only one cap is active are determined by
the frost properties and thermal inertia in the other hemisphere. During the times when both caps are active, they
not only both affect the pressure variation, they also
slightly affect each other through the dependence of frostpoint temperature on CO2 partial pressure. As an example, the rate and net amount of condensation in the south
is influenced by the mass of frost in the northern cap and
its rate of sublimation. Therefore, changing the properties
of one seasonal cap can result in a slightly different pressure variation even after it has disappeared. What this
implies is that the fits obtained by holding parameters in
one hemisphere constant, as in Fig. 6, are not quite as
good as they would have been if the parameters in both
hemispheres were allowed to vary. For example, Table 1
shows that a slightly better fit can be obtained for the
670/167 north/south thermal inertia combination than is
indicated in Fig. 6, where the albedo and emissivity values
in the south were fixed at their best-fit values for the 1340/
167 case.
D. Implications for Other Models
It is likely that more complicated models may also be
able to successfully fit observed Martian seasonal pressure variations using more physical processes and/or more
free parameters than the relatively simple model used in
this study. James and North (1982) showed that a twodimensional Mars climate model employing the radiative
effects of north polar hood clouds, dust clouds, and meridional heat transport could match the seasonal pressure
variations at the VL1 landing site, and then concluded
FITTING THE VIKING LANDER PRESSURE
11
CURVES
60
30
%
2
Ldm/dt
0
.%
5
Contour
Negative
-30
Edge
lntervol
10 (W/m’)
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Night
-
-
-----
-60
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0
30
60
90
120
150
160
210
240
270
300
330
360
90
60
30
:
.z
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-30
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Night
-
----
-60
-90
0
30
60
90
120
150
180
210
240
270
300
330
360
Ls
FIG. 9. (a) Contour plot of model-calculated daily average latent energy flux due to condensation or sublimation of CO2 using best-fit parameter
values corresponding to the 1340/167 north/south thermal inertia reference case, with a latitude resolution of 1”. (b) Contour plot of the quantities
in Fig. 9a multiplied by the corresponding area of each 1” latitude zone. The resulting values are in terawatts (TW).
that these factors may play an important role in determining the present behavior of the Martian seasonal polar
caps. Here, we show that an alternative model, which did
not include these factors, can successfully fit the same
observations with equal or better precision. In general,
then, the fact that a given model can fit the Viking pressure
curve does not, by itself, prove that any one particular
process it includes presently occurs on Mars. Furthermore, in our study, we also find that there are many
possible combinations of input parameter values that
could be used to fit the data. This implies that a fit to the
pressure curve, by itself, does not uniquely determine the
value of any single model input parameter. The accurate
reproduction of observed Martian seasonal pressure variations will continue to be an important challenge for future
modeling efforts, but the results of this study make it clear
that additional observational
constraints must also be
taken into account before unambiguous conclusions concerning the true properties and processes responsible for
the present CO2 cycle can be made.
E. Interpretation
of Derived Parameters
In their original paper, Leighton and Murray (1966)
admitted that their model “may constitute a significant
oversimplification of the actual situation on Mars.” Given
12
WOOD
AND
PAIGE
950
With Dust S t o r m s
900
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Oust Storms
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180
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FIG. 10. James and North's (1982) best fit to the VL1 data using a two-dimensional model that included the effects of global dust storms and
atmospheric heat transport and radiation, compared to the results of this same model using identical parameters except for the occurence of global
dust storms. (The dashed curve was not an attempt to fit VL data for a non-dust-storm year.)
that this model assumes no changes in input parameters
with time and ignores a number of potentially important
processes that could affect C02 frost behavior, such as
atmospheric radiation and energy transport--what then
can explain the apparent success of this model? One possibility is that atmospheric radiation and energy transport
have only minor net effects on the parts of the seasonal
polar caps that are most responsible for the seasonal pressure variations, which would imply that our derived frost
albedos and emissivities are close to their actual values.
Another possibility is that processes that are not considered in this model are indeed important, which would
imply that our derived frost albedos and emissivities may
differ significantly from their actual values. In this case,
our best-fit frost emissivities could include the net effects
of topography, polar hood clouds, or meridonal heat transport in either hemisphere, and our best-fit frost albedos
could include the net radiative effects of atmospheric
C02, or dust.
F. Temporal Variability
Another issue to consider is whether or not the properties and processes that determine the Viking Lander pressure curves themselves undergo significant seasonal variations. These could be caused by a multitude of factors,
including changes in frost and/or atmospheric radiative
properties with time, or by possible persistent changes in
the surface pressures at the Viking Lander sites due to
large-scale changes in the circulation of the Martian atmo-
sphere. The analysis by Hess et al. (1979) of the differences between the measured pressures at VL1 and VL2
during the first Viking year from Ls 120 to 300 indicated
that the hydrostatic effect can account for the major part
of the differences that are observed during this period,
but that additional, unexplained differences on the order
of 25 Pa may be present in the data. In our study, the
observed seasonal pressure variations at VL1 were assumed to be directly proportional to the mass of the atmosphere. All calculated seasonal pressure variations were
ultimately driven by seasonal variations in the distribution
of incident solar radiation, and no variations in frost properties with time were assumed. Our success in fitting the
seasonal components of the Viking pressure curve may
imply that the combined net effects of dynamically induced pressure variations and temporal changes in frost
radiative properties either worked in synchrony to cancel
each other out or were not important. The latter explanation seems more likely, since each of these should vary
independently with time.
The expected temporal variability of atmospheric effects is likely to also be evident on interannual timescales.
Global dust storms, for example, were observed to occur
during the first Viking year, but not during the second or
third. As discussed earlier, James and North (1982) were
able to obtain a good fit to the VL1 first year pressure data
by using a complex, two-dimensional model that included
the expected effects of global dust storms, atmospheric
radiation and heat transport, and polar hood phenomena.
Figure 10 shows the results of their best-fit model, as well
FITTING THE VIKING LANDER PRESSURE CURVES
as the results of their model using identical parameters
with the exception of dust storm effects (James and North
1982). As can be seen in the figure, James and North's
two-dimensional model appears to rely heavily on the
effects of the dust storm itself in order to fit the VL1 first
year data. To match the pressure minima around L S 360,
which is nearly the same in all 3 years, their model would
require a different set of atmospheric effects. One common aspect of the James and North best-fit model results
and our own is the need for a heat source in the north
polar region to delay the onset of frost condensation in
fall, and to reduce net frost condensation rates during fall
and winter. In our model, this heat source is not due to
dynamic and radiative processes occurring in the atmosphere, but to heat conduction from the subsurface. Of
the two, subsurface heat conduction would be expected
to exhibit the least interannual variability, and thus may
help to explain the interannual repeatability of the Viking
Lander seasonal pressure variations.
G. Potential Applications
The present CO 2 cycle on Mars is a complex and sensitive system. In many investigations of the atmosphere,
surface, and subsurface of Mars, it is desirable to have a
conceptually simple, computationally efficient, and physically based model that fits the Viking Lander pressure
curve and the seasonal polar cap retreat data. For problems that require knowledge of the mass of CO2 on the
surface, or the rates of CO2 condensation or sublimation,
at any given place and time, our model can provide estimates of these quantities that are consistent with available
observations. The model should also be useful for studying the effects of global dust storms, the long-term stability
of permanent polar caps, and the sensitivity of the seasonal cycle to variations in the orbital and axial elements
of Mars.
CONCLUSIONS
This investigation has shown that:
1. It is possible to fit the seasonal component of the
Viking Lander pressure variations to within a few pascal
using a diurnal and seasonal thermal model without any
explicit atmospheric contributions to the heat balance,
and parameters that are kept constant with time.
2. Although the best-fit model parameter values are not
unique, there is a fixed relationship between them, so that
if one is specified, the others are determined.
3. Subsurface heat conduction has significant effects on
Martian seasonal pressure variations, polar cap boundaries, and best-fit frost albedos and emissivities. It can
provide a means of influencing frost accumulation rates
that is highly repeatable from year to year.
13
4. Seasonal pressure variations are dominated by the
heat balance of the 55 ° to 70° latitude zone in both hemispheres. Thus, almost all of the most influential portions
of the seasonal polar caps receive some sunlight throughout the winter.
5. The albedo and emissivity values required to fit the
pressure curve with the model used in this study are not
consistent with the stability of permanent CO2 deposits at
either pole, even if the model is initialized with a cold
subsurface. This implies that the heat balance of the seasonal frost deposits at the south residual cap differs from
that of the most influential portions of the south seasonal
cap.
ACKNOWLEDGMENTS
We thank J. E. Tillman, the Viking Lander Meteorology Team, and
the Planetary Data System for supplying us with these invaluable data,
and P. B. James and R. W. Zurek for helpful reviews of this manuscript.
This work was supported by the NASA Mars Surface and Atmosphere
Through Time Program.
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